Dedicated to the memory of T. A. SpringerAbstract. Let V be a 2n-dimensional vector space over an algebraically closed field k with ch k = 2. Let G = GL(V ) and H = Sp 2n be the symplectic group obtained as H = G θ for an involution θ on G. We also denote by θ the induced involution on g = Lie G. Consider the variety G/H ×V on which H acts naturally. Let g −θ nil be the set of nilpotent elements in the −1 eigenspace of θ in g. The role of the unipotent variety for G in our setup is played by g −θ nil × V , which coincides with Kato's exotic nilpotent cone. Kato established, in the case where k = C, the Springer correspondence between the set of irreducible representations of the Weyl group of type C n and the set of H-orbits in g −θ nil × V by applying Ginzburg theory for affine Hecke algebras. In this paper we develop a theory of character sheaves on G/H × V , and give an alternate proof for Kato's result on the Springer correspondence based on the theory of character sheaves.

Résumé. -Soient G un groupe algébrique réductif connexe défini sur Fq et F l'endomorphisme de Frobenius correspondant. Soit σ un automorphisme rationnel quasicentral de G. Nous construisons ci-dessous l'équivalent des représentations de GelfandGraev du groupe G F = G F · σ , lorsque σ est unipotent et lorsqu'il est semi-simple. Nous montrons de plus que ces représentations vérifient des propriétés semblables a celles vérifiées par les représentations de Gelfand-Graev dans le cas connexe en particulier par rapport auxéléments réguliers. Abstract (Regular Elements and Gelfand-Graev Representations for Disconnected Reductive Groups)Let G be a connected reductive group defined over Fq and let F be the corresponding Frobenius endomorphism. Let σ be a quasi-central automorphism of G, which means that σ is quasi-semi-simple (i.e. σ stabilises (T ⊂ B) where T is a maximal torus included in a Borel subgroup B of G) and dim(G σ ) > dim(G σ ) for any quasi-semisimple automorphism σ = σ • ad(g), where ad(g) is the conjugation by g for all g ∈ G.We suppose also that σ is rational.We define in this article Gelfand-Graev representations for the group G F = G F · σ when σ is unipotent and when it is semi-simple, which extend the σ-stable GelfandGraev representations for connected reductive groups.

This paper is the second part of the papers in the same title. In this paper, we prove a conjecture of Achar-Henderson, which asserts that the Poincaré polynomials of the intersection cohomology complex associated to the closure of Sp 2n -orbits in the Kato's exotic nilpotent cone coincide with the modified Kostka polynomials indexed by double partitions, introduced by the first author. Actually this conjecture was recently proved by Kato by a different method. Our approach is based on the theory of character sheaves on the exotic symmetric space.